Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 3959−3968
pubs.acs.org/JPCL
Randomness-Induced Phonon Localization in Graphene Heat Conduction Shiqian Hu,†,‡,§,○ Zhongwei Zhang,†,‡,§,○ Pengfei Jiang,†,‡,§ Jie Chen,*,†,‡,§ Sebastian Volz,*,‡,∥,⊥ Masahiro Nomura,# and Baowen Li*,▽
J. Phys. Chem. Lett. 2018.9:3959-3968. Downloaded from pubs.acs.org by UNIV OF SUNDERLAND on 10/18/18. For personal use only.
†
Center for Phononics and Thermal Energy Science, School of Physics Science and Engineering, and Institute for Advanced Study, Tongji University, Shanghai 200092, People’s Republic of China ‡ China−EU Joint Lab for Nanophononics, School of Physics Science and Engineering, Tongji University, Shanghai 200092, People’s Republic of China § Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, People’s Republic of China ∥ Laboratoire d’Energétique Moléculaire et Macroscopique, Combustion UPR CNRS 288, Ecole Centrale Paris, Grande Voie des Vignes, F-92295 Chatenay-Malabry, France ⊥ Laboratory for Integrated Micro and Mechatronic Systems, CNRS-IIS UMI 2820, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan # Institute of Industrial Science, The University of Tokyo, Meguro-ku, Tokyo 153-8505, Japan ▽ Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309, United States S Supporting Information *
ABSTRACT: Through nonequilibrium molecular dynamics simulations, we report the direct numerical evidence of the coherent phonons participating in thermal transport at room temperature in graphene phononic crystal (GPnC) structure and evaluate their contribution to thermal conductivity based on the two-phonon model. With decreasing period length in GPnC, the transition from the incoherent to coherent phonon transport is clearly observed. When a random perturbation to the positions of holes is introduced in a graphene sheet, the phonon wave-packet simulation reveals the presence of notable localization of coherent phonons, leading to the significant reduction of thermal conductivity and suppressed length dependence. Finally, the effects of period length and temperature on the coherent phonon contribution to thermal conductivity are also discussed. Our work establishes a deep understanding of the coherent phonons transport behavior in periodic phononic structures, which provides effective guidance for engineering thermal transport based on a new path via phonon localization.
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phonon interaction. In this case, the coherence/correlation of phonons is lost, that is, the incoherent effect. Traditionally, the thermal conductivity is manipulated through impurities,1,2 nanoparticles,3−6 defects,7−12 ion-intercalation,13 encapsulation,14 interparticle constriction,15 and static electricity16 in materials by manipulating the incoherent phonon scattering.17,18 Recently, a new strategy focuses on managing phonons by superlattice and nanostructured phononic crystals (PnCs),4,19−28 which allows for controlling heat by engineering the phonon band structures (coherent phonon transport).29−33 The coherent phonon transport behavior has been observed in several theoretical34−36 and experimental37−39 investigations on superlattice structures, such as in Si/Ge,34 graphene/boron nitride,35,36 and GaAs/
n recent years, efficient modulation of the thermal transport has become more and more urgent due to the imperative need to enhance the heat dissipation rate in the continuously miniaturized electronic devices and the acute demand to improve energy conversion efficiency based on thermoelectrics. Because phonons dominate heat transfer in crystalline dielectrics and semiconductors, the manipulation of the thermal conductivity (κ) can be realized through controlling phonons by two types of mechanisms: the incoherent phonon scattering mechanism and the coherent mechanism. Here the coherent mechanism means the manipulation of phonons via the modification of phonon dispersion arising from the phonon wave effects, such as the opening of phononic band gaps and the reduction of group velocity, which reflects the coherent nature of phonons. In contrast, the incoherent mechanism means to treat phonon scatterings similar to the particle collisions that originated from the diffusive phonon scatterings by the boundary, impurity, or intrinsic anharmonic phonon− © 2018 American Chemical Society
Received: May 28, 2018 Accepted: July 3, 2018 Published: July 3, 2018 3959
DOI: 10.1021/acs.jpclett.8b01653 J. Phys. Chem. Lett. 2018, 9, 3959−3968
Letter
The Journal of Physical Chemistry Letters AlAs28 superlattice. Moreover, the manipulation of the coherent phonons in PnCs that have minimal influence on electrons is considered to be the key for improving thermoelectric efficiency.19 Although extensive efforts have been made to investigate the reduction of κ by constructing PnC structures and coherent phonon transport has been predicted by theoretical studies,40,41 only a few experimental works25,38 could directly observe coherent phonon transport in PnCs due to the difficulty of avoiding the phonon phase-breaking events caused by phonon−phonon and phonon−roughness interactions. In most experiments, κ was found to increase monotonically with the increase in the lattice period, and the measured42−44 κ values of PnC samples are significantly lower than those of theoretical predictions after taking into account the combination of material removal and incoherent phonon scattering. The role played by coherent and incoherent phonons in PnC has not yet been systematically investigated and remains unclear, which requires further investigation to separate these two types of phonons and quantify their contributions to thermal transport. In this work, using nonequilibrium molecular dynamics simulations (NEMD), we report the direct evidence of the coherent phonon in periodic atomic structure and further quantify the contributions of coherent and incoherent phonons to the total thermal conductivity of graphene phononic crystals (GPnC) via the construction of a disordered graphene phononic crystal (DGPnC) structure. First, the signature for coherent phonon thermal transport is observed in GPnC structures by varying the period length. Furthermore, we find that the thermal conductivity of DGPnC is significantly reduced compared with that of GPnC, showing negligible length dependence for the thermal conductivity of DGPnC. On the basis of the two-phonon model,45 we then reveal the two different transport behaviors of coherent phonons in GPnC and DGPnC. On the basis of the spatial distribution of pressure and the phonon wave-packet simulation, the direct numerical evidence of the emergence of the coherent phonons is obtained. The underlying physical mechanism is uncovered by analyzing the spatial energy distribution of the localized phonons and the phonon transmission. Finally, the physical parameters affecting coherent phonon thermal transport are studied. A clear picture of the coherent phonons transport properties in GPnC is obtained in this work. As shown in Figure 1a, the GPnC structure is composed of the holey unit cells constructed by creating holes in the pristine graphene, simultaneously ensuring that the hole edge is zigzag, simply to remove the effect of different defect types on the thermal transport of graphene. 11 The unit cells are characterized by the period L0 and porosity. Here the porosity is defined as NR/NP, where NR and NP are the number of removed atoms and the total number of atoms in the pristine graphene, respectively (see Supporting Information Part I). Thermal conductivity of graphene is, in principle, composed of electron and phonon contributions. Before we start our systematic investigation, we first estimate the electron contribution to the total thermal conductivity in GPnC structure by using the nonequilibrium Green’s function method (see Supporting Information Part II for calculation details). Our calculation results (Figure S2 in the Supporting Information) show that the electron contributes only ∼13% of the total thermal conductance in the undoped GPnC structure at room temperature, which is in good agreement with the
Figure 1. Effects of porosity, period, and temperature on the formation of coherent phonons. (a) Schematic picture of the GPnC. Fixed boundary conditions are used along the length (L) direction, whereas periodic boundary conditions are used along the width (W) direction. The GPnC structure is characterized by the period (L0) and the porosity. (b) κGPnC as a function of the period L0 with different porosities at 300 K. The system length L is fixed at 82 nm. (c) Temperature effect on κGPnC with a fixed porosity of 25%.
calculation result from previous theoretical study on graphene.46 Therefore, we only consider the lattice thermal conductivity in the following discussion. NEMD simulations in this paper are performed by using LAMMPS package.47 Brenner48 potential is used to simulate the covalent bonding interaction between carbon atoms. Fixed and periodic boundary conditions are adopted along the length and width directions, respectively. To establish a temperature gradient, two Langevin thermostats49 with different temperatures are applied to the two ends of the simulation system. The thermal conductivity κ is calculated based on Fourier’s Law J κ=− (1) ∇T where ∇T and J are, respectively, the temperature gradient and the heat flux. Here the effective thermal conductivity for the finite-size system is obtained through eq 1, as the thermal transport behavior in our study is not necessarily diffusive. A similar treatment has been widely adopted in the literature studies50−54 on the anomalous heat conduction in lowdimensional lattices and materials. More simulation details can be found in the Supporting Information Part III. We first study the thermal conductivity of GPnCs at 300 K. The width of GPnC is chosen as 8 nm, which is large enough to eliminate the numerical size effect stemming from the periodic boundary conditions set in the width direction. The system length L is fixed to 82 nm. Figure 1b shows the thermal conductivity, κ, of GPnCs as a function of the period L0 for different porosities at 300 K. The results show that κ first decreases and then increases with decreasing L0. Using another empirical force field for carbon (the optimized Tersoff55 potential), we have verified that this nonmonotonic depend3960
DOI: 10.1021/acs.jpclett.8b01653 J. Phys. Chem. Lett. 2018, 9, 3959−3968
Letter
The Journal of Physical Chemistry Letters ence of κ on L0 is generic (see Supporting Information Figure S4), regardless of the specific force field used. The critical period, at which the minimum κ is located, can be regarded as the dividing line between the incoherent-phonon-dominated regime and the coherent-phonon-dominated regime. This crossover phenomenon of the incoherent and coherent regimes was verified experimentally in a superlattice structure.38 Above the critical period, large L0 provides enough space (higher probability) for the occurrence of phonon−phonon scatterings before a phonon traverses the holes in each unit cell, which can prevent the formation of coherent phonon modes. In this case, incoherent phonons dominate the thermal transport in the GPnCs. As a result, with a fixed total length and decreasing L0, the number of unit cells (holes) increases, leading to the stronger edge scatterings and the decrease in the thermal conductivity of the GPnCs. In contrast, as L0 further decreases below the critical period, more phonons can be transmitted ballistically before getting reflected at the holes edge, which is beneficial to form coherent phonons due to the phonon interference caused by the multiple reflections at the periodic holes edge. In this coherent region, a smaller L0 corresponds to a larger Brillouin zone, leading to less band folding in the phonon dispersion. This results in a weaker flattening of the phonon band (see Supporting Information Figure S5), and thus leads to the increase in the phonon group velocity, which is responsible for the increase in the thermal conductivity.56,57 Moreover, at the same period L0, κGPnC is significantly reduced when increasing porosity from 10 to 25% due to the stronger edge scattering (Figure 1b), which is consistent with previous works.24,32 Figure 1c shows κ of GPnCs versus L0 at a fixed porosity of 25% with the temperature varying from 300 to 1000 K. One interesting finding is that the nonmonotonic behavior and the κ minimum become less distinct at elevated temperature. This is because higher temperature causes stronger anharmonic phonon−phonon scattering, leading to the suppression of the coherent phonons. As shown in Figure 1b, the length scale for the critical period is very small, on the order of a few nanometers at room temperature. Defects are inevitably introduced during the experimental fabrication of such a fine PnC structure, which may affect the observation of such coherent phonon transport in real experiment. Furthermore, because the incoherent phonons only depend on the period L0 and the porosity, whereas the coherent phonons depend additionally on the exact periodicity, that is, the distribution of these holes in unit cells,42 the introduction of randomness provides us an opportunity to distinguish the contributions from the coherent and incoherent phonons in the thermal conductivity of GPnC, as studied in 2D phononic crystal and disordered nanostructures.25 In this regard, we construct the DGPnC structure (Figure 2a) by randomly distributing the position of the hole in each unit cell (see Supporting Information Part V) while keeping the period L0 and porosity the same as that in GPnC. Figure 2b shows the length dependence of thermal conductivity for both GPnC (κGPnC) and DGPnC (κDGPnC) with the same porosity of 10% at 300 K. Here L0 is fixed at 1.24 nm, which falls into the coherent phonons dominated regime. NEMD simulation results show that κGPnC (rectangle) is always higher than κDGPnC (triangle) for the same length, except for the sufficiently short length (L < 10 nm), which suggests that the boundary scattering is the dominant
Figure 2. Schematic of DGPnC and length dependence of thermal conductivity in GPnC and DGPnC at 300 K. (a) Schematic of DGPnC structure. (b) Rectangles and triangles denote the simulations result of the thermal conductivity of GPnC and DGPnC, respectively. On the basis of the model fitting, the dasheddotted line draws the incoherent phonon contribution to thermal conductivity, which is the same for both GPnC and DGPnC. The dashed line stands for the coherent phonon contribution to thermal conductivity of DGPnC. The dashed-dotted-dotted line denotes the coherent phonon contribution to thermal conductivity of GPnC. The solid line draws the ballistic thermal conductivity of GPnC from NEGF calculations. The periodic length and porosity are fixed at 1.24 nm and 10%, respectively. The error bars are the standard deviations of three independent simulations with different initial conditions.
mechanism in determining thermal conductivity on a very small length scale so that the ballistic phonon transport takes place. To verify this hypothesis, we further calculated the ballistic thermal conductivity of GPnC at room temperature by using the nonequilibrium Green’s function (NEGF) method combined with the Landauer formalism (see Supporting Information Part II for details), where the anharmonic phonon−phonon interaction is completely neglected in the calculations. As shown in Figure 2b, the ballistic thermal conductivity from NEGF calculations (solid line) agrees very well with the thermal conductivity result from NEMD simulation (rectangle) when the total length, L, is sufficiently short (
